Japan is facing its greatest crisis since the second world war, its prime minister, Naoto Kan, warned as the country struggled for a third day to avert a nuclear disaster following the massive earthquake and tsunami.

Police warn that the death toll could exceed 10,000 in Miyagi prefecture alone, and a UN agency reported that more than 1,600 people were confirmed dead. Most are thought to have drowned.

"The earthquake, tsunami and the nuclear incident have been the biggest crisis Japan has encountered in the 65 years since the end of the second world war," Kan told news media. "We're under scrutiny on whether we, the Japanese people, can overcome this crisis."

The extent of this challenge was underscored by the news that the capital of the world's third largest economy will see rolling three-hour power cuts from Monday.

Officials have ordered the measure because nuclear plant closures have drastically reduced the electricity supply. They warn that the blackouts could last for weeks.

"We have to avoid at all costs a sudden power shortage whose scale could have devastating consequences for the economy and people's lives," Kan said.

Friday's 9.0 magnitude earthquake, upgraded from 8.9, is the worst in Japan's recorded history and the fifth strongest worldwide in the past century. It caused a tsunami 10 metres high in some places.

As aftershocks continued to shake the coast, the Foreign Office urged Britons to avoid all non-essential travel to Tokyo and the north-east of Japan, while the US state department recommended that Americans stay away from Japan.

The Japanese meteorological agency warned that there was a 70% possibility of a magnitude 7 or greater tremor during the next three days. It lifted the tsunami warning, but cautioned that aftershocks could cause further waves.

The agency also warned that a volcano in the south was erupting again after a fortnight without noticeable activity. Ash and rock spewed from the crater of Shinmoedake on Kyushu island, 950 miles from the epicentre of Friday's tremor. There were no reports of injuries or damage.

Tokyo has doubled the number of troops in its rescue and relief team, but damage to roads and bridges has hampered efforts.

"Rescue and relief operations are being hampered by continuous aftershocks, tsunami alerts and fires. Many areas along the north-east coast remain isolated and unreachable," the UN Office for the Co-ordination of Humanitarian Affairs said in a statement.

It reported that at least 1,600 people were known to have died and 10,000 were missing. Almost 600,000 have been displaced, or evacuated, because of the earthquake and tsunami or radiation fears, it added.

Etsuko Oyama, who was rescued by a neighbour after the wave swept her 400 metres from her home, struggled to hold back tears as she spoke to NHK's news channel, the Japanese public broadcaster.

"I grabbed my daughter's hand but I lost my grip when I was swept away by the debris and water," she said. "I managed to survive but my daughter was washed away … I hope she is still alive somewhere."

Japanese troops rescued a 60-year-old man who was swept 10 miles out to sea on the roof of his home. But Hiromitsu Shinkawa said his wife had been washed away when the tsunami hit as they returned home to gather possessions after the quake.

While there were welcome moments of celebration as survivors were reunited, any good news was generally alloyed by grief.

With phone services still down in many areas, many survivors made their way to civic centres to check boards with names of others known to have survived and those who had died.

The Kyodo news agency reported that more than 20,000 buildings nationwide were either destroyed or badly damaged. Some 2.6 million households were without power and 1.4 million without drinking water, said Japan's ministry of health, labour and welfare, although electricity was restored to some parts of Sendai city. Others in the area reported food shortages.

Survivors huddled in public shelters for another night in near-freezing temperatures. In addition to the central government's relief efforts, cities and private businesses have sent help. The deputy mayor of Tokyo said the city was dispatching 384,000 blankets and 9,000 portable toilets, while Nissin, an instant noodle manufacturer, said it would send a million packets and "kitchen cars" equipped with stoves and water.

International relief and rescue teams from more than a dozen countries, including China, the US and the UK, are arriving to help Japanese troops. Almost 70 countries have offered donations, such as India's consignment of blankets and self-contained field hospitals from Australia.

Many of the donors, including Pakistan and Sri Lanka, have been scarred by earthquakes or tsunamis in recent years.

Kandahar in southern Afghanistan also pledged cash. "I know $50,000 [£31,000] is not a lot of money for a country like Japan, but it is a show of appreciation from the Kandahar people," Ghulam Haidar Hamidi, the mayor, told Reuters.

Yukio Edano, Japan's chief cabinet secretary, has said the government would use its contingency funds of some 200bn yen (£1.5bn) to pay for the relief effort. But there is already talk of a possible temporary tax increase to fund relief work.

Japan, overtaken by China last year as the world's second largest economy, will require tens of billions of dollars for reconstruction. The 1995 Kobe earthquake is said to have been the most expensive natural disaster in history, causing more than £1bn damage.

The country will also need to count the cost of industrial losses, with much of the north-east at a standstill.